Why do power lines sag when they are heated up? I was reading some information about the 2003 power blackout in the Northeastern US. 

Beginning early in the afternoon of August 14, 2003 big transmission lines began to fail in First Energy's operating area, several because the utility had not kept up with tree-trimming, so that as heavily loaded lines heated up, they sagged into brush and shorted out. As one went down, the next would become too loaded, sag still more, and short, and so on. All that, the result of a serious infringement of operating standards and no small matter in its own right,
  would have remained a local problem if First Energy and the midwestern power regulator had quickly recognized what was going on and had promptly cut service to enough customers to keep the whole system from getting overloaded. 

From "A Coal Power Ban Is a Necessity." Coal. Ed. Michael Logan. 5 Aug. 2010
What about overheating causes power lines to sag? I understand that increasing the load on the line will increase the line's temperature due to the resistance in the cables. Why does this increased heat cause the lines to sag though?
 A: Besides the thermal expansion noted elsewhere (which is correct), heat causes a different kind of failure in power lines. Heat anneals the aluminum used, causing it to loose strength and eventually fail.
The standard cables used are called ACSR (Aluminum Conductor Steel Reinforced) and the strength of the cable is shared between the steel core and the aluminum strands. Above 93°C the aluminum used (AL1350-H19) looses its heat treatment and it's breaking strength goes down. Eventually with some extra wind, or other loading conditions it might fail. Possibly the lower strength lowers the resistance to fatigue and thus aeolian vibrations might make the cable fail. More reading here and here.
Also what happens sometimes is at the ends of the conductor there are connectors that carry the current and those may fail if operated at high temperatures (runaway heat condition) as their sealing compound inside "melts" away allowing for corrosion to enter in the the steel supports of the connectors. 
These have sparked the need for high temperature conductors such as ACSS, ACCR and others. Southwire has some nice write up like here. Operating temperatures can be 120°C or over 200°C with some of the more exotic (and expensive) cables used today.
Also, high operating temperatures accelerate creep, which is a gradual expansion of the cable with time. As creep progresses the cable will sag more and it might impact a tree or structure (bad). The effect is not linear, and the higher the temperature creep moves much more quickly and can become the dominant effect of sag within a few years.
Maybe additionally the increased sag increases the swept area of the conductor (becomes longer between the supports) making it more susceptible to wind and ice loading.
PS. I used to work in the transmission line industry and your question is very welcome here.
A: The conductor material (copper, aluminum, whatever) expands when heated. When the temperature increases, the length of the power line between two towers increases due to thermal expansion, and the line sags because of the increased slack.
A: There is thermal expansion of metals, and in a power line the length will become greater than the installation one. Gravity takes up the slack and lowers the lowest point of the line. The shape taken by suspended strings/cables  with only their own weight is called a catenary, and the length depends on the constants of the solution ( alternative analysis in the catenary link). 
A: During summer due to heat, the intra molecular space increases and the substance expands. As the substance expands, the weight of the conductor increases resulting in the increased sag.
